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Lesson: Matrix Differentiation

Worksheet • 5 Questions

Q1:

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

  • A
  • B
  • C
  • D
  • E

Q2:

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

  • A
  • B
  • C
  • D
  • E

Q3:

Consider the vector space of infinitely differentiable functions. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation 𝐴 = 𝐷 + 2 𝐷 + 1 2 .

  • A 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 + 𝐢 𝑒 1 𝑑 2 𝑑
  • B 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 βˆ’ 𝐢 𝑒 1 𝑑 2 𝑑
  • C 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 1 𝑑
  • D 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 1 βˆ’ 𝑑
  • E 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 + 𝐢 𝑒 1 βˆ’ 𝑑 2 βˆ’ 𝑑

Q4:

Consider the vector space of infinitely differentiable functions. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation 𝐴 = 𝐷 + 5 𝐷 + 4 2 .

  • A 𝑦 ( 𝑑 ) = 𝐢 𝑒 + 𝐢 𝑒 1 βˆ’ 𝑑 2 βˆ’ 4 𝑑
  • B 𝑦 ( 𝑑 ) = 𝐢 𝑒 βˆ’ 𝐢 𝑒 1 𝑑 2 4 𝑑
  • C 𝑦 ( 𝑑 ) = 𝐢 𝑒 1 βˆ’ 𝑑
  • D 𝑦 ( 𝑑 ) = 𝐢 𝑒 1 𝑑
  • E 𝑦 ( 𝑑 ) = 𝐢 𝑒 + 𝐢 𝑒 1 𝑑 2 4 𝑑

Q5:

Apply the linear differential operator 𝐷 to evaluate the following expression: ο€Ή 𝐷 βˆ’ 2 𝐷 + 4  ο€Ή π‘₯ 𝑒 + 5 π‘₯ + 2  2 π‘₯ 2 .

  • A 3 π‘₯ 𝑒 + 1 8 βˆ’ 2 0 π‘₯ + 2 0 π‘₯ π‘₯ 2
  • B 3 π‘₯ 𝑒 βˆ’ 1 8 + 2 0 π‘₯ βˆ’ 2 0 π‘₯ π‘₯ 2
  • C π‘₯ 𝑒 + 1 0 π‘₯ π‘₯
  • D π‘₯ 𝑒 + 2 5 π‘₯ π‘₯ 2
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